9.66 (1984)

a) B. Jonsson’s Conjecture: Elementary equivalence is preserved under taking free products in the class of all groups, that is, if $\operatorname{Th}(G_1) = \operatorname{Th}(G_2)$ and $\operatorname{Th}(H_1) = \operatorname{Th}(H_2)$ for groups $G_1, G_2, H_1, H_2$, then $\operatorname{Th}(G_1 * H_1) = \operatorname{Th}(G_2 * H_2)$.

b) It may be interesting to consider also the following weakened conjecture: if $\operatorname{Th}(G_1) = \operatorname{Th}(G_2)$ and $\operatorname{Th}(H_1) = \operatorname{Th}(H_2)$ for countable groups $G_1, G_2, H_1, H_2$, then for any numerations of the groups $G_i$, $H_i$ there are $m$-reducibility $T_1 \equiv_m T_2$ and Turing reducibility $T_1 \equiv_T T_2$, where $T_i$ is the set of numbers of all theorems in $\operatorname{Th}(G_i * H_i)$. A proof of this weakened conjecture would be a good illustration of application of the reducibility theory to solving concrete mathematical problems.